Abstract
By using the concept of Frechet differentiability of mapping, we present the Kuhn-Tucker optimality conditions for weakly efficient solution, Henig efficient solution, superefficient solution, and globally efficient solution to the vector equilibrium problems with constraints.
Highlights
Some authors have studied the optimality conditions for vector variational inequalities
Vector variational inequality problems and vector optimization problems, as well as several other problems, are special realizations of vector equilibrium problems see 20, 21 ; it is important to give the optimality conditions for the solution to the vector equilibrium problems for in this way we can turn the vector equilibrium problem with constraints to a corresponding scalar optimization problem without constraints, and we can determine if the solution of the scalar optimization problem is a solution of the original vector equilibrium problem
Under the assumption of convexity, Gong 22 investigated optimality conditions for weakly efficient solutions, Henig solutions, superefficient solutions, Journal of Inequalities and Applications and globally efficient solutions to vector equilibrium problems with constraints and obtained that the weakly efficient solutions, Henig efficient solutions, globally efficient solutions, and superefficient solutions to vector equilibrium problems with constraints are equivalent to solution of corresponding scalar optimization problems without constraints, respectively
Summary
Some authors have studied the optimality conditions for vector variational inequalities. Some authors have derived the optimality conditions for weakly efficient solutions to vector optimization problems see 4–19. Qiu 23 presented the necessary and sufficient conditions for globally efficient solution under generalized cone-subconvexlikeness. Gong and Xiong 24 weakened the convexity assumptions in 22 and obtained necessary and sufficient conditions for weakly efficient solution, too. By using the concept of Frechet differentiability of mapping, we study the optimality conditions for weakly efficient solutions, Henig solutions, superefficient solutions, and globally efficient solutions to the vector equilibrium problems. We give Kuhn-Tucker necessary conditions to the vector equilibrium problems without convexity conditions and Kuhn-Tucker sufficient conditions with convexity conditions
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